On Sublinear Inequalities for Mixed Integer Conic

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On Sublinear Inequalities for Mixed Integer Conic Programs Fatma Kılın¸ c-Karzan · Daniel E. Steffy

Submitted: December 2014; Revised: July 7, 2015

Abstract This paper studies K-sublinear inequalities, a class of inequalities with strong relations to K-minimal inequalities for disjunctive conic sets. We establish a stronger result on the sufficiency of K-sublinear inequalities. That is, we show that when K is the nonnegative orthant or the secondorder cone, K-sublinear inequalities together with the original conic constraint are always sufficient for the closed convex hull description of the associated disjunctive conic set. When K is the nonnegative orthant, K-sublinear inequalities are tightly connected to functions that generate cuts—so

called cut-generating functions. As a consequence of the sufficiency of Rn + -sublinear inequalities, we also provide an alternate and straightforward proof of the sufficiency of cut-generating functions for mixed integer linear programs, a result recently established by Cornu´ejols, Wolsey and Yıldız.

1 Introduction

In this paper, we consider disjunctive conic sets of the form S (A, K, B) := {x ∈ E : Ax ∈ B, x ∈ K},

where K ⊂ E is a regular cone (full-dimensional, closed, convex, and pointed) in a finite dimensional Euclidean space (E, h·, ·i) with inner product h·, ·i, A : E → Rm is a linear map, and ∅ = 6 B ⊆ Rm is a set of right hand side vectors. Important examples of regular cones include the nonnegative q n n n 2 orthant R+ , the second-order cone L := {x ∈ R : xn ≥ x1 + . . . + x2n−1 }, and the positive n semidefinite cone S+ := {x ∈ Rn×n : aT x a ≥ 0 ∀a ∈ Rn }. We note that we make no structural assumptions on B, in particular, B may be either finite or infinite. Without loss of generality we assume that B 6= ∅. We are interested in studying the structure of valid linear inequalities defining the closed convex hull of S (A, K, B), denoted by conv(S (A, K, B)). Note that the cases S (A, K, B) = ∅ or conv(S (A, K, B)) = K are trivial. Hence, throughout the paper we often restrict our attention to the interesting cases, where conv(S (A, K, B)) is nonempty and not equal to K. The disjunctive conic sets S (A, K, B) can represent multi-term disjunctions on K and thus naturally arise in relaxations for Mixed Integer Conic Programs (MICPs) with a regular cone K. These

F. Kılın¸c-Karzan Tepper School of Business, Carnegie Mellon University, Pittsburgh, PA, 15213, USA. E-mail: [email protected] D. E. Steffy Department of Mathematics and Statistics, Oakland University, Rochester, MI, 48309 USA. E-mail: [email protected]

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sets are instrumental in derivation and analysis of general cutting planes for MICPs (see [21]). For example, such sets cover the simpler setups commonly studied such as two-term disjunctions or split disjunctions. The particular case of two-term or split disjunctions on K = Ln has recently attracted a lot of attention [1, 2, 6, 7, 9, 12, 15, 22, 23, 24, 25, 29]. n When K = Rn + and B is a finite set, Johnson [20] initiated the study of S (A, R+ , B ) and intron duced and characterized R+ -minimal valid linear inequalities for it. Prior to this, minimal inequalities have been studied in a number of setups related to Mixed Integer Linear Programs (MILPs). For example, in the context of group problem, a characterization of minimal inequalities via the so-called subadditivity, symmetry, and periodicity conditions was given in [16, 19]. Jeroslow [18] extended this characterization from group theoretic framework to arbitrary MILPs under the assumption that the feasible region of the MILP is bounded. In particular, an explicit characterization of minimal inequalities based on the value functions of MILPs was given in [18]. Blair [10] broadened the results from [18] to the case where the entries of the constraint matrix in MILP are rational. Bachem et al. [3] further enhanced these characterizations by introducing b-complementarity condition for a master group problem induced by an MILP. We refer the readers to [17] for a survey on the literature for subadditive approach to MILP. The developments in these papers are mainly based on the value functions of MILPs and therefore have strong connections to the subadditive strong duality theory for MILPs. This body of earlier work, with the exception of [20], specifically focus on feasible sets of MILPs in a finite dimensional setup [10, 17, 18, 20] or the associated infinite relaxations [3, 16, 17, 19]. As a result, the framework of these papers are related to disjunctive sets S (A, Rn + , B ) with specific B . More recently, [5, 11] has also studied minimal valid inequalities but in the context of infinite relaxations associated with MILPs. As opposed to the earlier literature, [5, 11] studied more general sets, where the set B is composed of general lattice points [11] or lattice points contained in a rational polyhedron [5] and established strong connections between the minimal inequalities and the gauge functions of maximal lattice-free sets. Kılın¸c-Karzan [21] followed up on the general and flexible disjunctive framework of [20] by generalizing the study of S (A, K, B) to general regular cones K and removing the finiteness assumption on B. In particular, [21] introduces K-minimal inequalities for disjunctive conic sets and studies their structure in detail in terms of their existence, sufficiency, necessary conditions, and sufficient conditions for K-minimality. It is shown in [21] that minimality notion of an inequality is a natural result of identifying dominance relations among valid linear inequalities; and therefore, it should be based on the smallest regular cone K that can be used to describe S (A, K, B). This point is important even in the case of a disjunctive set associated with an MILP (see [21, Section 2.2 and Remark 7]); yet it has been overlooked in all of the papers in this literature. Based on the necessary conditions for K-minimality, K-sublinear inequalities, which contain K-minimal inequalities as a subclass and have easier to characterize algebraic properties, were also introduced in [21]. The purpose of this paper is to answer open questions surrounding K-sublinear inequalities, their sufficiency status in important specific cases, and resulting implications. Previously, in [21], it was established that whenever there is at least one K-minimal inequality, K-minimal inequalities together with the original x ∈ K constraint are sufficient for describing conv(S (A, K, B)). However, the existence of K-minimal inequalities depends on certain structural properties of S (A, K, B); and it is possible to have nontrivial and interesting sets S (A, K, B) for which none of the describing inequalities are K-minimal (see [21, Example 6]). Despite this, and the fact [21, Theorem 1] that K-sublinear inequalities subsumes K-minimal inequalities, the question of whether K-sublinear inequalities would be sufficient in general (independent of any structural assumption on S (A, K, B)) was left open in [21]. In this paper, we first establish a stronger connection between K-sublinear inequalities and extreme inequalities. That is, every extreme inequality is either K-sublinear or can be captured by the conic constraint x ∈ K along with all valid equations. On a related note, we also show that not every extreme inequality is K-sublinear or captured by the constraint x ∈ K. This demonstrates that there can be valid equations which are neither K-sublinear or implied by the constraint x ∈ K. Nevertheless, we provide a positive answer to the open question on the sufficiency

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n of K-sublinear inequalities in two important cases of interest, namely when K = Rn + or K = L . In n n the case of K = R+ , by studying the connection between R+ -sublinear inequalities and the functions that generate cut coefficients—so called cut-generating functions (CGFs)—we also provide an alternative and simplified proof of a recent result by Cornuejols et al. [14] on the sufficiency of CGFs. The second part of the paper considers the case when K = Ln . In this case, we show that Ln -sublinearity definition can be considerably simplified. As a consequence, we also establish that any valid inequality for S (A, Ln , B) with n ≥ 3 is always Ln -sublinear. The remainder of the paper is organized as follows. Section 2 introduces our notation and describes previous results. Section 3 studies some properties of the K-sublinear inequalities for general K. Section 4 presents our results on the sufficiency of Rn + -sublinear inequalities along with x ≥ 0 constraint. In this section, we also discuss the connections with CGFs and the implications of the sufficiency of Rn + -sublinear inequalities on the sufficiency of CGFs. Finally, Section 5 examines K-sublinear inequalities for second-order cones K = Ln .

2 Notation and Preliminaries

We start by introducing our notation. For a set S , we denote its topological interior with int(S ), its closure with S and its boundary with ∂S = S \ int(S ). We use conv(S ) to denote the convex hull of S , conv(S ) for its closed convex hull, and cone(S ) to denote the cone generated by the set S . We define the kernel of a linear map A : E → Rm as Ker(A) := {u ∈ E : Au = 0} and its image as Im(A) := {Au : u ∈ E}. We use A∗ to denote the conjugate linear map1) given by the identity y T Ax = hA∗ y, xi ∀(x ∈ E, y ∈ Rm ).

For a given cone K ⊂ E , we let Ext(K) denote the set of its extreme rays, and K∗ to denote its n n dual cone given by K∗ := {y ∈ E : hx, yi ≥ 0 ∀x ∈ K} . Note that the cones Rn + , L , and S+ are all ∗ self-dual, i.e., K = K; and, in the first two cases, the corresponding Euclidean space E is just Rn with dot product as the corresponding inner product. Throughout the paper, we use Matlab notation to denote vectors and matrices and all vectors are to be understood in column form. We will use ei for the ith unit vector of Rn . We use notation a K b to denote the relation a − b ∈ K, that is, the conic inequality between vectors a and b.

2.1 Classes of Valid Linear Inequalities Given S (A, K, B), we are interested in the valid linear inequalities for conv(S (A, K, B)). Consider the set of all vectors 0 6= µ ∈ E such that ϑ(µ) defined as ϑ(µ) := inf {hµ, xi : x ∈ S (A, K, B)} x

(1)

is finite. Then any nonzero vector µ ∈ E and a number µ0 ≤ ϑ(µ) gives a valid linear inequality of the form hµ, xi ≥ µ0 for S (A, K, B). We denote the corresponding valid inequality by (µ; µ0 ). We say that a valid inequality (µ; µ0 ) is tight 2) if µ0 = ϑ(µ). If both (µ; µ0 ) and (−µ; −µ0 ) are valid inequalities, then hµ, xi = µ0 holds for all x ∈ S (A, K, B); and in this case, we refer to (µ; µ0 ) as a valid equation for S (A, K, B). For any vector δ ∈ K∗ \{0}, the inequality (δ ; 0) is always valid for S (A, K, B). We refer to these inequalities as cone-implied inequalities. We define C (A, K, B) = {(µ; µ0 ) ∈ E × R : µ0 ≤ ϑ(µ)}, 1 ) Note that when E = Rn , and a linear map A : Rn → Rm is just an m×n real-valued matrix, and its conjugate is given by its transpose, i.e., A∗ = AT . 2 ) We note that our definition of tightness of an inequality does not require the corresponding hyperplane to have a nonempty intersection with the feasible region, as is sometimes the definition used in the literature.

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the convex cone of all valid linear inequalities. The constraint x ∈ K captures all of the coneimplied inequalities. In order to understand the linear inequalities necessary for the description of conv(S (A, K, B)), we study the generators of C (A, K, B). Note that any convex cone K can be written as the sum of linear subspace L and a pointed cone C . Here L represents the largest linear subspace contained in the cone K , also referred to as the lineality space of K . A unique representation of K in the form of K = L + C can be obtained by requiring that C is contained in the orthogonal complement of L. A generating set (GL , GC ) for a cone K is defined to be a minimal set of elements GL ⊆ L, GC ⊆ C such that     X X αw w + λv v : λv ≥ 0 . K=   w∈GL

v∈GC

For a generating set (GL , GC ) of C (A, K, B) we refer to the vectors in GL as generating valid equalities and the vectors in GC as generating valid inequalities of C (A, K, B). Without loss of generality, we can assume that all of the vectors in GL are orthogonal to each other; and each vector in GC is orthogonal to all vectors in GL (see [21, Remark 2]). An inequality (µ; µ0 ) ∈ C (A, K, B) is called an extreme inequality if there exists a generating set for C (A, K, B) including (µ; µ0 ) as a generating inequality either in GL or in GC . Given two vectors, u, v ∈ K where K is a cone with lineality space L, u is said to be an L-multiple of v if u = αv + ` for some α > 0, and ` ∈ L. From this definition, it is clear that if u is an L-multiple of v , then v is also an L-multiple of u. Thus, whenever the lineality space L of the cone C (A, K, B) is nontrivial, i.e., L 6= {0}, the generating valid inequalities are only defined uniquely up to the L-multiples. Understanding the structure of extreme valid linear inequalities is critical in terms of understanding the structure of conv(S (A, K, B)). On the other hand, characterizing all extreme valid inequalities can be quite difficult for an arbitrary set S (A, K, B). A middle ground is obtained by studying the structure of slightly larger classes of inequalities. In particular, classes of minimal and sublinear inequalities, where these notions are defined with respect to the cone K, had received quite some interest in the recent years. For example, such a study on the structure of tight, Kminimal inequalities underlies the developments of nonlinear disjunctive cuts in [22, 23] for the case of two-term linear disjunctions applied to a second-order cone. The cone K in the description of S (A, K, B) plays a critical role in identifying dominance relations among valid linear inequalities. Consider two valid linear inequalities for S (A, K, B) given by (µ; µ0 ) and (ρ; ρ0 ) such that ρ K∗ µ and ρ 6= µ. Then we say that (ρ; ρ0 ) dominates (µ; µ0 ) whenever ρ0 ≥ µ0 . In fact, when (ρ; ρ0 ) dominates (µ; µ0 ), we have µ = ρ + δ for some δ ∈ K∗ \ {0} and ρ0 ≥ µ0 . Moreover, the inequality (ρ; ρ0 ) together with the conic constraint x ∈ K imply the inequality (µ; µ0 ) because hµ, xi = hρ + (µ − ρ), xi = hρ, xi + hµ − ρ, xi ≥ ρ0 ≥ µ0 ,

| {z } ≥ρ0

|

{z

≥0

}

where the first inequality follows from x ∈ K and µ − ρ = δ ∈ K∗ . In the case of K = Rn + and associated finite and infinite relaxations for MILPs, minimality of a valid inequality is traditionally defined with respect to K = Rn + . That is, a valid inequality (µ; µ0 ) is minimal if for all ρ ≤ µ and ρ 6= µ, the inequality given by (ρ; µ0 ) is not valid (see [20]). This means reducing any µi for i ∈ {1, . . . , n} in a minimal inequality (µ; µ0 ) will lead to a strict reduction in its right hand side value.3 The following natural extension of the minimality definition to disjunctive conic sets, and thus to the MICP case, was introduced in [21]: 3 We note that the valid inequalities that are referred as minimal in [3, 4, 10, 18] correspond to tight and Rn + minimal inequalities with respect to the definitions in this paper.

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Definition 1 (K-minimal inequality) A valid linear inequality (µ; µ0 ) for S (A, K, B) with µ 6= 0 and µ0 ∈ R is K-minimal if for all valid inequalities (ρ; ρ0 ) for S (A, K, B) satisfying ρ 6= µ, and ρ K∗ µ, we have ρ0 < µ0 .

In particular, the definition of K-minimality implies that if (µ; µ0 ) with µ 6= 0 is valid but not K-minimal then there exists another valid inequality (ρ; ρ0 ) that dominates it. That is, (µ; µ0 ) is a consequence of the inequality (ρ; ρ0 ) and the conic constraint x ∈ K. Note that such dominance relations are of great interest in obtaining smaller yet sufficient sets of valid linear inequalities. Therefore, the selection of cone K plays a critical role in the description of S (A, K, B). For further discussion on the topic, we refer the reader to [21, Remarks 5 and 7]. For a given S (A, K, B), we denote the set of all K-minimal inequalities by Cm (A, K, B). It was shown in [21, Proposition 1] (see [20] for the case of K = Rn + ) that Cm (A, K, B ) 6= ∅ only if the following assumption holds: Assumption 1: For each δ ∈ K∗ \ {0}, there exists some xδ ∈ S (A, K, B) such that hδ, xδ i > 0.

Moreover, by [21, Proposition 2 and Corollary 2], under Assumption 1, K-minimal inequalities together with the original x ∈ K constraint, are sufficient to describe conv(S (A, K, B)). This motivated further study of the properties of K-minimal inequalities. By isolating a number of algebraic necessary conditions for K-minimality, [21] suggested the following class of K-sublinear inequalities: Definition 2 (K-sublinear inequality) Given S (A, K, B), a linear inequality (µ; µ0 ) with µ 6= 0 and µ0 ∈ R is K-sublinear if it satisfies the conditions (A.1(α)) for all α ∈ Ext(K∗ ) and (A.2) where

(A.1(α)) 0 ≤ hµ, ui for all u s.t. Au = 0 and hα, viu + v ∈ K ∀v ∈ Ext(K), (A.2) µ0 ≤ hµ, xi for all x ∈ S (A, K, B). When an inequality satisfies (A.1(α)) for all α ∈ Ext(K∗ ), we say that it satisfies condition (A.1). In this definition, while condition (A.2) is simply true for every valid inequality, condition (A.1) is not very intuitive. A particular and simple case of (A.1) was identified in [21]; and it is of interest: Any valid inequality (µ; µ0 ) satisfying (A.1) also satisfies the condition (A.0) 0 ≤ hµ, ui for all u ∈ K such that Au = 0. Note that condition (A.0) is in fact equivalent to µ ∈ (K ∩ Ker(A))∗ = K∗ + (Ker(A))∗ = K∗ + Im(A∗ ),

where the last equation follows from the facts that (Ker(A))∗ = Ker(A)⊥ = Im(A∗ ) and K∗ +Im(A∗ ) is closed whenever K is closed [28, Corollary 16.4.2]. In fact, it is known [21, Proposition 6] that conditions (A.0) and (A.2) are satisfied by any valid inequality (µ; µ0 ) with µ 6= 0 and µ0 ∈ R. Therefore, condition (A.1) is the main condition defining the interesting structure of K-sublinear inequalities. When K = Rn + , conditions (A.0)-(A.2) underlie the definition of subadditive inequalities from [20] (see e.g., [21, Remark 9]). We next have the following immediate corollary regarding specific K-sublinear inequalities. Corollary 1 Any valid inequality (µ; µ0 ) with µ ∈ Im(A∗ ) is K-sublinear. Proof Condition (A.2) is satisfied since (µ; µ0 ) is valid. Because µ ∈ Im(A∗ ), using the fact that all of the u’s in condition (A.1) are from Ker(A), we arrive at hµ, ui = 0, thus µ satisfies condition (A.1) as well. u t

Let Cs (A, K, B) denote the cone of K-sublinear inequalities. Conditions (A.1)-(A.2) imply that Cs (A, K, B) is indeed a convex cone. By [21, Theorem 1], we have Cm (A, K, B) ⊆ Cs (A, K, B) ⊆ C (A, K, B). Based on the sufficiency of K-minimal inequalities, this automatically implies that under Assumption 1, the valid inequalities from Cs (A, K, B) together with x ∈ K are sufficient to describe conv(S (A, K, B)). Nonetheless, the question of whether Assumption 1 is needed in this sufficiency result for K-sublinear inequalities was not studied in [21]. We pursue this question in the next section.

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3 Relation between K-sublinear Inequalities and Extreme Inequalities

In this section, for general regular cones K, we study the connection between K-sublinear inequalities and extreme inequalities describing conv(S (A, K, B)). Without relying on Assumption 1, we first establish a stronger structural result between K-sublinear inequalities and extreme inequalities. The following lemma from [20] is useful in our analysis: Lemma 1 Let K = L + C where C is a pointed cone and L is the largest linear subspace contained in C. Suppose v is in a generating set for cone K and there exists v 1 , v 2 ∈ K such that v = v 1 + v 2 , then v 1 , v 2 are L-multiples of v. Proposition 1 Every extreme inequality for S (A, K, B) is either K-sublinear or an L-multiple of a cone-implied inequality. Proof Suppose (µ; µ0 ) is an extreme inequality. Let (GC , GL ) be a generating set for C (A, K, B) containing (µ; µ0 ). We will show that if (µ; µ0 ) is not K-sublinear, then it is an L-multiple of a cone-implied inequality. Assume that (µ; µ0 ) is valid but not K-sublinear. Then it violates condition (A.1) and hence for some α ∈ Ext(K∗ ), there exists u satisfying hµ, ui < 0 such that Au = 0 and hα, viu + v ∈ K ∀v ∈ Ext(K) holds. Define the linear map Z : E → E by Zx = hx, uiα + x

for any x ∈ E.

We first show that (Zµ; µ0 ) is a valid inequality. Since hZµ, xi = hµ, Z ∗ xi ∀x ∈ E , it is enough to show that Z ∗ x ∈ S (A, K, B) for any x ∈ S (A, K, B); then the validity of (µ; µ0 ) will give hZµ, xi = hµ, Z ∗ xi ≥ µ0 . Because A : E → Rm , its conjugate A∗ : Rm → E . Without loss of generality let A∗ ei =: Ai ∈ E for i = 1, . . . , m, where ei is the ith unit vector in Rm . Then from u ∈ Ker(A), we have hAi , ui = 0, which implies ZA∗ ei = hAi , ui + Ai = Ai for all i = 1, . . . , m. Therefore, we arrive at ZA∗ = A∗ . Moreover, because A : E → Rm and Z : E → E are linear maps, ZA∗ is a linear map and its conjugate is given by AZ ∗ = A. Consider any x ∈ S (A, K, B), then we have Ax ∈ B and x ∈ K. Hence, AZ ∗ x = Ax ∈ B. Also, because u satisfies hα, viu + v ∈ K ∀v ∈ Ext(K), we get hα, xiu + x ∈ K holds for all x ∈ K. Thus, for all w ∈ K∗ , we have hw, Z ∗ xi = hZw, xi = h(hw, uiα + w), xi = hw, uihα, xi+hw, xi = hw, hα, xiu + xi ≥ 0. | {z } ∈K

Therefore, Z ∗ x ∈ K, proving that for any x ∈ S (A, K, B), we have Z ∗ x ∈ S (A, K, B). Additionally, (µ−Zµ) = −hµ, uiα ∈ K∗ since hµ, ui < 0. So (µ−Zµ; 0) is a cone-implied inequality. Finally, µ = Zµ + (µ − Zµ) making the original inequality a conic combination of (Zµ; µ0 ) and the cone-implied inequality (µ − Zµ; 0). Since we have assumed that (µ; µ0 ) is extreme, by Lemma 1, these inequalities must be L-multiples of each other, implying that (µ; µ0 ) is an L-multiple of a cone-implied inequality. u t Consider S (A, K, B) and an inequality (µ; µ0 ) for it. We associate with µ the following set Dµ := {λ ∈ Rm : A∗ λ K ∗ µ},

(2)

and its support function given by σDµ (z ) := supλ∈Rm {hz, λi : λ ∈ Dµ }. These entities play critical roles in the characterization of K-sublinear inequalities. We now show that as long as non-trivial valid linear inequalities for S (A, K, B) exist, K-sublinear inequalities must exist as well. Proposition 2 Suppose conv(S (A, K, B)) 6= K. Then, K-sublinear inequalities always exist.

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Proof Because conv(S (A, K, B)) 6= K, there exists at least one extreme inequality (µ; µ0 ) that is noncone-implied. Then by Proposition 1, it is either K-sublinear or an L-multiple of a cone-implied inequality. If it is K-sublinear, then we are done. So, suppose it is an L-multiple of a cone-implied inequality, that is (µ; µ0 ) = (ρ; ρ0 ) + (δ ; 0) where (ρ; ρ0 ) is a valid equation and δ ∈ K∗ \ {0}. Since (µ; µ0 ) is non-cone-implied, we have (ρ; ρ0 ) is also non-cone-implied. Because (ρ; ρ0 ) is a valid equation, both (ρ; ρ0 ) and (−ρ; −ρ0 ) satisfy condition (A.0). Thus, ρ ∈ Im(A∗ ) + K∗ and −ρ ∈ Im(A∗ ) + K∗ . Then there exists λ+ , λ− ∈ Rm \ {0} and δ + , δ − ∈ K∗ such that ρ = A∗ λ+ + δ + = −A∗ λ− − δ − . Therefore, −A∗ (λ+ + λ− ) = δ + + δ − implying δ + + δ − ∈ Im(A∗ ) ∩ K∗ . If Im(A∗ ) ∩ K∗ = {0}, then δ + = −δ − . Because K∗ is pointed and both δ + = −δ − ∈ K∗ and δ − ∈ K∗ , we have δ + = δ − = 0. Then ρ ∈ Im(A∗ ); and using Corollary 1, we conclude that (ρ; ρ0 ) is K-sublinear. When Im(A∗ ) ∩ K∗ 6= {0}, there exists 0 6= ξ ∈ Im(A∗ ) ∩ K∗ . For that ξ , we have 0 ∈ Dξ , where the set Dξ is defined as in (2). Thus, its support function σDξ (·) is nonnegative, i.e., σDξ (b) ≥ 0 for all b. From [21, Proposition 8], we have ϑ(ξ ) = inf b∈B σDξ (b). Then using inf b∈B σDξ (b) ≥ 0 and by Corollary 1, this leads to (ξ ; 0) is K-sublinear. u t

Proposition 1 suggests that a detailed study of the structure of valid equations for S (A, K, B) is key to understanding the sufficiency of K-sublinear inequalities. In order to completely argue for the sufficiency of K-sublinear inequalities for general regular cones K, one may conjecture that every extreme valid equation is either K-sublinear or cone-implied. Unfortunately, this is not true as we demonstrate with a counter example in the next section. 4 Sufficiency of K-sublinear Inequalities for K = Rn +

We first note that due to the decomposable structure of Rn + , following [21, Remark 9], the conditions (A.1(α)) required for Rn -sublinearity simplify and become (A.0) and the following conditions + (A.1i) for i = 1, . . . , n: (A.1i) 0 ≤ µT u for all u such that Au = 0 and u + ei ∈ Rn +, where ei denotes the ith unit vector in Rn . T In addition, for a given µ ∈ Rn , µ satisfies condition (A.0) if and only if µ ∈ Rn + + Im(A ), and thus, Dµ 6= ∅. We also recall the following result [21, Proposition 8]: n T Remark 1 Consider S (A, Rn + , B ) and µ ∈ R+ + Im(A ). Then ϑ(µ) = inf b∈B σDµ (b).

♦

Furthermore, Theorem 4 and Proposition 10 from [21] are handy in our developments (see also [20, Theorems 9-10] and [21, Remarks 9, 10 and 11]). We summarize these here: n T Theorem 1 Consider S (A, Rn + , B ) and µ ∈ R+ + Im(A ). Let Ai denote the i-th column of the matrix n A. Then (µ; µ0 ) is an R+ -sublinear inequality if and only if it is valid (µ0 ≤ ϑ(µ)) and σDµ (Ai ) = µi for all i = 1, . . . , n.

We start by considering an illustrative example of an extreme inequality that is not Rn +sublinear. Then we derive a Rn + -sublinear inequality that dominates it, using methods that are later on generalized. Example 1 Consider the set S (A, K, B) given by



   1 0 −1 1 A= , B= and K = R3+ . 01 1 0 Note that S (A, K, B) = {[1; 0; 0]}. A generating set (GC ; GL ) for the corresponding cone of valid inequalities C (A, K, B) is given by GC = ∅ and GL = {(e1 ; 1), (e2 ; 0), (e3 ; 0)}. Consider the valid

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Fatma Kılın¸c-Karzan, Daniel E. Steffy

inequality (µ; µ0 ) = (−e3 ; 0) ∈ GL . We will show that this extreme inequality is not R3+ -sublinear. Given (µ; µ0 ) = (−e3 ; 0), let us examine condition (A.1i) for i = 2, which requires u2 ≥ 0 for all u such that Au = 0 and u + e2 ∈ R3+ . Yet, the vector u = [1; −1; 1] ∈ Ker(A) ∩ (−e2 + R3+ ) violates this condition, implying that (µ; µ0 ) = (−e3 ; 0) is a valid inequality that is not R3+ -sublinear. Let us now construct an R3+ -sublinear inequality that dominates (µ; µ0 ) = (−e3 ; 0). Given µ, we consider the set Dµ = {λ ∈ R2 : A∗ λ ≤ µ} = {λ ∈ R2 : λ1 ≤ 0, λ2 ≤ 0, −λ1 + λ2 ≤ −1}. Using the support function of Dµ , i.e., σDµ (b) = supλ {bT λ : A∗ λ ≤ µ}, we construct a new vector based on the value of this function for each column Ai of the matrix A, that is, η = [σDµ (A1 ); σDµ (A2 ); σDµ (A3 )] = [0; −1; −1]. The validity of (η ; µ0 ) with µ0 = 1 is easily seen by the fact that S (A, K, B) = {[1; 0; 0]}. Moreover, Dη = {λ ∈ R2 : A∗ λ ≤ η} = {λ ∈ R2 : λ1 ≤ 0, λ2 ≤ −1, −λ1 + λ2 ≤ −1} = Dµ . Thus, ηi = σDη (Ai ) for all i = 1, 2, 3. Then using Theorem 1, we conclude that (η ; µ0 ) with µ0 = 0 is an R3+ -sublinear inequality. Finally, note that µ = η + δ with δ = e2 ∈ R3+ ; and thus (η ; µ0 ) dominates (µ; µ0 ). ♦ We next state our main result of this section which establishes that Rn + -sublinear inequalities along with the constraint x ≥ 0 are sufficient to describe conv(S (A, Rn + , B )). n Proposition 3 Let K = Rn + . Then any valid inequality (µ; µ0 ) for S (A, R+ , B ) is equivalent to or n dominated by an R+ -sublinear inequality (where the domination is defined with respect to K = Rn + ). m Proof Given a valid inequality (µ; µ0 ) for S (A, Rn : AT λ ≤ + , B ), let us consider the set Dµ = {λ ∈ R µ}, and its support function σDµ (·). Based on these, we will generate a new vector η by setting ηi = σDµ (Ai ) for all i = 1, . . . , n where Ai is the i-th column of the matrix A. First, note that because (µ; µ0 ) is a valid inequality, it satisfies condition (A.0). Then by [21, Theorem 3], we have Dµ = 6 ∅ and ηi = σDµ (Ai ) ≤ µi . Thus, µ ≥ η . Moreover, since x ∈ Rn + and σDµ (·) is a support function and thus it is positively homogeneous and subadditive, for any b, we conclude

X  n o n n σ ( A ) x = inf inf η T x : Ax = b, x ∈ Rn : Ax = b, x ∈ R i i + + Dµ x x | {z } i=1 =σDµ (Ai xi )

 n X n Ai xi ) : Ax = b, x ∈ R+ ≥ σDµ (b). ≥ inf σDµ ( 

x

i=1

| {z } =b

Also, from the validity of (µ; µ0 ) we have ϑ(µ) ≥ µ0 . Then by Remark 1, we conclude ϑ(µ) = inf b∈B σDµ (b) ≥ µ0 . This implies n o T n inf {η T x : x ∈ S (A, Rn + , B )} = inf inf η x : Ax = b, x ∈ R+ ≥ inf σDµ (b) = ϑ(µ) ≥ µ0 . x

b∈B x

b∈B

Hence, (η ; µ0 ) is a valid inequality that dominates (µ; µ0 ). In order to finish the proof, we need to show that (η ; µ0 ) ∈ Cs (A, K, B). To see this, let us consider the set Dη and its support function σDη (·). Once again using [21, Theorem 3], we have Dη 6= ∅ and σDη (Ai ) ≤ ηi for all i. Besides, from the definition of η , ηi = σDµ (Ai ); and hence Dµ ⊆ {λ ∈ Rm : AT i λ ≤ ηi } holds for any i. Thus, we reach to Dµ ⊆ Dη . Using the monotonicity of the support functions, we then obtain σDµ (a) ≤ σDη (a) for all a ∈ Rm . In particular, ηi = σDµ (Ai ) ≤ σDη (Ai ) for all i. Therefore, we conclude ηi = σDη (Ai ) for all i. Then Theorem 1 implies that (η ; µ0 ) is an Rn u t + -sublinear inequality. A few remarks are in order. The sufficiency result given in Proposition 3 leads to nice consequences. When K = Rn + , based on the precise correspondence (see [21, Remark 9]) between subaddin tive inequalities of Johnson [20] and Rn + -sublinear inequalities, and the fact that every R+ -sublinear

On Sublinear Inequalities for Mixed Integer Conic Programs

9

inequality is generated by the support function of a certain set (see Theorem 1 and Remark 1), we conclude that the support functions of the nonempty sets of form Dµ = {λ ∈ Rm : A∗ λ ≤ µ} are sufficient to generate all valid inequalities for conv(S (A, Rn + , B )). As a result of this, we define support functions of nonempty sets D ⊂ Rm as relaxed cut-generating functions. m Definition 3 Given S (A, Rn + , B ) and a set ∅ 6= D ⊂ R , we say that the support function σD : m R → (R ∪ +∞) of D is a relaxed cut-generating function for S (A, Rn + , B ). T Note that for fixed n, A and B, and a µ ∈ Rn + +Im(A ), the relaxed CGF associated with Dµ 6= ∅ n leads to a valid inequality for S (A, R+ , B) via Remark 1. At first sight, the relaxed CGFs might seem to depend on all of the data A, n and B, m associated with the specific instance S (A, Rn + , B) and therefore work only for the given S (A, Rn + , B ). We next show that due to the sublinearity of relaxed CGFs, their validity is indeed independent of0 A and n; and thus, they can be used to generate valid inequalities for other instances S (A0 , Rn + , B ) as long as the set B is kept the same.

Proposition 4 Suppose B ⊂ Rm is given. Let σ (·) be a relaxed CGF for S (A, Rn + , B ) associated with a nonempty set D ⊂ Rm . Then, the inequality 0

n

0

0

m×n

x ∈ S (A , R+ , B) where A ∈ R th column of the matrix A0 .

0

Pn0

0 i=1 σ (Ai )xi 0 0

≥ inf b∈B σ (b) is valid for any

, the dimension n and A are arbitrary, and A0i denotes the i-

Proof Because σ (·) is a relaxed CGF, it is a support function; and thus, it is sublinear (subadditive

P n0 Pn0 0 0 0 and convex). Then for any x ∈ S (A0 , Rn + , B ), we have i=1 σ (Ai )xi ≥ σ ( i=1 Ai xi ) = σ (b) for some b ∈ B, where the inequality is due to the subadditivity of σ (·) and the equation is due to P n0 0 0 0 n0 x ∈ S (A0 , Rn u t + , B ). Hence, i=1 σ (Ai )xi ≥ inf b∈B σ (b) holds for all x ∈ S (A , R+ , B ). While relatively simple, Proposition 4 has an important consequence. Together with Remark 1, Proposition 4 indicates that the relaxed CGFs σDµ obtained from Rn + -sublinear inequalities (µ; µ0 ) P n0 n for S (A, R+ , B) can be used to generate cuts of the form i=1 σDµ (A0i )xi ≥ inf b∈B σDµ (b) ≥ µ0 for 0

0 0 any set S (A0 , Rn + , B ) where A and n are arbitrary, i.e., they can be taken as varying. We now turn our attention to cuts of the form cT x ≥ 1 separating origin from conv(S (A, Rn + , B )). This setup naturally arises in the context of separating a fractional solution from the feasible region of an MILP [13] and has attracted some attention in the recent literature. In particular, some of the previous literature, including [13, 14], have specifically focused on the separation of the origin from conv(S (A, Rn + , B )) under the assumption that B is closed and 0 6∈ B . In this case, 0∈ / conv(S (A, Rn + , B )), which ensures the existence of valid inequalities separating the origin from conv(S (A, Rn + , B )). Our analysis to follow does not depend on the closedness of B ; and thus we will not make this assumption. Instead, we simply assume that 0 ∈ / conv(S (A, Rn + , B )). Proposition 3 implies that for any cut cT x ≥ 1 separating the origin from conv(S (A, Rn , B )), we either have + (c; 1) ∈ Cs (A, K, B) or there will be at least one Rn -sublinear inequality that dominates (c; 1). + Therefore, by focusing on the associated relaxed CGFs, we arrive at the following corollary of Proposition 3.

Corollary 2 Let Ai be the i-th column of the matrix A for all i = 1, . . . , n. Consider the case with m T K = Rn and 0 ∈ / conv(S (A, Rn x ≥ 1 separating the origin +, B ⊂ R + , B )). Then any valid inequality cP n n from conv(S (A, R+ , B)) is equivalent to or dominated by one of the form i=1 σ (Ai )xi ≥ 1, obtained m from a relaxed CGF σ : R → (R ∪ +∞). Proof Since (c; 1) separates the origin from conv(S (A, Rn + , B )), any valid inequality that is equivalent n to or dominates (c; 1) will also separate the origin from conv(S (A, Rn + , B )). Because R+ -sublinear n inequalities along with x ≥ 0 are sufficient to describe conv(S (A, R+ , B)) (see Proposition 3), and the cone-implied inequalities xi ≥ 0 do not separate the origin, there exists (µ; µ0 ) ∈ Cs (A, K, B) that is equivalent to or dominates (c; 1). Thus, µ0 ≥ 1. Finally, from Theorem 1, we have, for every Rn + -sublinear inequality (µ; µ0 ), the support function σDµ of the associated nonempty set Dµ generates the coefficients of this inequality, i.e., σDµ (Ai ) = µi for all i. u t

10

Fatma Kılın¸c-Karzan, Daniel E. Steffy

m Recently, Conforti et al. [13] studied a variant of the set S (A, Rn is a fixed + , B ) where B ∈ R m×n nonempty and closed set such that 0 6∈ B, yet n and A ∈ R are varying. It is easy to see [13, Lemma 2.1] that in such a setup, 0 6∈ conv(S (A, Rn + , B )). For this particular setup, the authors are interested in generating cuts that separate the origin from conv(S (A, Rn + , B )) and introduce the concept of a cut-generating function as follows:

Definition 4 Given a nonempty and closed set B ∈ Rm satisfying 0 6∈ B, a cut-generating function for B is a function f : Rn → RP such that for any natural number n ∈ N and any matrix A ∈ Rm×n , n the linear inequality given by n i=1 f (Ai )xi ≥ 1 is valid for S (A, R+ , B ) where Ai is the i-th column of the matrix A.

In this setup, various properties of cut-generating functions and their relations to lattice-free sets were studied in [13]. One question left open in [13] was whether or not CGFs are sufficient to give all of the cuts separating the origin from conv(S (A, Rn + , B )). In the follow up work [14], by using the value functions of linear programs and arguments from convex analysis, a positive answer to this question was provided under an additional structural assumption. From Remark 1, Proposition 4, and Corollary 2, we immediately deduce a related result. That is, without any further technical assumptions, the relaxed CGFs are always sufficient to generate all cuts separating the origin from conv(S (A, Rn + , B )) and A and n can be taken as varying exactly as in the framework of [13]. In contrast to this, Example 6.1 of [13] shows that there are sets of the form S (A, Rn + , B ) such that CGFs are not sufficient to generate all cuts separating the origin. This disparity between the sufficiency of relaxed CGFs and the insufficiency of regular CGFs is due to the fact that regular CGFs are required to be finite valued everywhere while relaxed CGFs are not. In particular, the relaxed CGFs used in Proposition 4 and Corollary 2 are simply support functions of certain sets; and therefore they may not be finite valued everywhere. We note that it is not necessary to require a function to be finite valued everywhere in order to use it to generate cuts for a given problem instance S (A, Rn + , B ). As long as the function is finite valued on the set of columns of A, it will lead to nontrivial valid inequalities. Nevertheless, for a function to work (i.e. generate nontrivial valid inequalities) for all instances of the form S (A, Rn + , B ) with varying A and n, the finite valuedness of the function is important. Within the framework of [13], the sufficiency of CGFs was established in [14] under the additional structural assumption that B ⊆ cone(A). We next show that under the same assumption, we can in fact guarantee finiteness of relaxed CGFs as well. T Proposition 5 Suppose 0 6∈ conv(S (A, Rn + , B )) and B ⊆ cone(A). Let c x ≥ 1 be a valid inequality n separating the origin from conv(S (A, R+ , B)). Then

(a) Dc := {λ ∈ Rm : A∗ λ ≤ c} is nonempty and the support function of Dc leads to a valid inequality that is equivalent to or dominates cT x ≥ 1. (b) Let V denote the set of extreme points of the polyhedral set Dc , and ρ be such that ρ ≥ max maxv∈V kvk∞ , 1 + inf λ∈Dc kλk∞ . Then Dc,ρ := {λ ∈ Rm : A∗ λ ≤ c, kλk∞ ≤ ρ} is nonempty. Moreover, the support function of Dc,ρ is finite valued everywhere, piecewise linear, and leads to a valid inequality that is equivalent to or dominates cT x ≥ 1. Proof (a) This part immediately follows from Corollary 2 and Proposition 3. (b) Let B (0, ρ) := {λ ∈ Rm : kλk∞ ≤ ρ}, where ρ ≥ 1 is as defined above. For any v ∈ V , using the definition of ρ, we have v ∈ B (0, ρ) as well. Since Dc,ρ = Dc ∩ B (0, ρ), we get v ∈ Dc,ρ for any v ∈ V , which also proves that Dc,ρ is nonempty whenever V = 6 ∅. Also, if V = ∅, then ρ = 1 + inf λ∈Dc kλk∞ = 1 + minλ∈Dc kλk∞ because Dc is nonempty and polyhedral. Hence, ¯ ∈ Dc such that kλk ¯ ∞ ≤ ρ; thus Dc,ρ 6= ∅ in this case as well. Because Dc,ρ is there exists λ a nonempty bounded set, its support function is finite valued everywhere. Also, Dc,ρ ⊆ Dc implies σDc,ρ (z ) ≤ σDc (z ) for every z ∈ Rn . Therefore, σDc,ρ (Ai ) ≤ σDc (Ai ) ≤ ci where the last inequality follows from part (a) and Ai denote the i-th column of the matrix A. To finish the proof, we need to show that the cut generated using σDc,ρ (·) with a right hand side value of 1 is valid (this will also show that this cut is equivalent to or dominates cT x ≥ 1). Note

On Sublinear Inequalities for Mixed Integer Conic Programs

11

that, for any z ∈ Rn such that σDc (z ) is finite, we have σDc (z ) = maxv∈V {z T v} ≤ σDc,ρ (z ) ≤ σDc (z ), which implies σDc (z ) = σDc,ρ (z ). Also, the set of recessive directions of Dc are given by Rec(Dc ) = {d ∈ Rn : A∗ d ≤ 0}. So, σDc (z ) = +∞ if and only if there exists 0 6= d ∈ Rec(Dc ) such that dT z > 0. Next, we show that under the assumption that B ⊆ cone(A), for every b ∈ B, we have dT b ≤ 0 for all d ∈ Rec(Dc ), implying σDc (b) < +∞. Because B ⊆ cone(A), ∗ n for any b ∈ B, ∃xb ∈ Rn + such that b = Axb . Also, d ∈ Rec(Dc ) if and only if A d ∈ R− . Then ∗
T T T xb = d Axb = d b, which implies that σDc (b) ≤ 0 < +∞. Hence, σDc (b) = σDc,ρ (b) 0≥ A d | {z } |{z} ∈ Rn −

∈Rn +

for all b ∈ B. Thus, we have inf b∈B σDc,ρ (b) = inf b∈B σDc (b) ≥ 1, where the last inequality follows from part P(a). Therefore, using Proposition 4, the relaxed CGF σDc,ρ (·) leads to the valid inequality n i=1 σDc,ρ (Ai )xi ≥ 1. Moreover, because σDc,ρ (·) leads to an inequality which is equivalent to or dominates (c; 1), this concludes the proof. u t Proposition 5 establishes that the relaxed CGF σDc,ρ (·) is actually finite valued. This taken together with Proposition 4, we conclude that σDc,ρ (·) is a CGF in the usual sense. Therefore, Proposition 5 recovers the main result, Theorem 1.1, of [14]. Note that when we ignore the restriction of a CGF being finite valued, then the existence of an Rn + -sublinear inequality equivalent to or dominating (c; 1), and hence a relaxed CGF leading to the corresponding Rn + -sublinear inequality, i.e., Corollary 2, is independent of the condition B ⊆ cone(A); and in the proof of Proposition 5, the condition B ⊆ cone(A) is used only to ensure the finiteness of the support function (relaxed CGF). Remark 2 The sufficiency of CGFs for S (A, Rn + , B ) has strong connections with the subadditive

(superadditive) duality theory for MILPs as well. The feasible set of any MILP problem can be converted into the form of optimizing a linear function over S (A, Rn + , B ) by appropriately selecting A and B. Specifically, we can write the set of feasible solutions of an MILP in disjunctive conic form as follows:      ¯b + Rm ¯ A n + ¯ ¯ {x ∈ Rn x∈ . + : Ax ≥ b, xi ∈ Z+ ∀i = 1, . . . , `} = x ∈ R+ : I` 0 Z`+ |{z} | {z } | {z } :=K :=A

:=B

The subadditive (superadditive) strong duality theorem for MILPs states that when there is at least one feasible solution, that is, S (A, Rn + , B ) 6= ∅, then one can write a dual optimization problem over all functions f that are nondecreasing with respect to Rm + and subadditive (superadditive); and this dual attains the same objective value as the primal MILP problem (see [19, 17, 27]). Indeed, the functions appearing in the strong dual formulation of MILPs are CGFs because these functions act locally on each individual variable by only considering the associated data Ai and producing the cut coefficient ci . As a result, strong MILP duality theorem is equivalent to the sufficiency of these functions for the corresponding specific sets S (A, Rn + , B ) associated with the set of feasible solutions of MILPs. Recently, under a technical condition, Moran et al. [26] has extended this strong duality theory to MICPs of a specific form (see [26, Theorem 2.4]). The structure of MICPs studied in [26] leads to specific disjunctive conic sets S (A, K, B) where the domination is defined with respect to a nonnegative orthant and the cone involved in MICP appears in the definition of the set B (see [21, Example 3]). We refer the reader to [21, Remark 12] for additional discussion relating the work of [26] to CGFs. ♦

5 Characterization of K-sublinear Inequalities for K = Ln

In this section, we refine the K-sublinearity conditions (A.0)-(A.2) for the case of second-order cone K = Ln . In particular, for Ln with n ≥ 3, we show that condition (A.1) is equivalent to condition

12

Fatma Kılın¸c-Karzan, Daniel E. Steffy

(A.0); and hence one does not need to verify condition (A.1) explicitly. One of the consequences of this is the sufficiency of Ln -sublinear inequalities along with x ∈ Ln constraint for describing conv(S (A, Ln , B)). Note that K = L2 is just a rotation of the nonnegative orthant which can be covered by the analysis in the previous section. In the case of K = Ln for a fixed n and a given α ∈ Ext(Ln ), we define the following sets for convenience in our analysis: ( ) n− X1 2 n T 2 Vα = v ∈ R : α v = 1, vn = vi , vn ≥ 0 and i=1



n

n



Xα = x ∈ R : x ∈ L − v ∀v ∈ Vα .

The following lemma gives an alternative condition that can be used to verify condition (A.1(α)). Lemma 2 Consider S (A, Ln , B). Let α ∈ Ext(Ln ). Then the linear inequality (µ; µ0 ) satisfies

0 ≤ hµ, ui

for all u ∈ Ker(A) ∩ Xα ,

(3)

only if it satisfies condition (A.1(α)). Proof Let u be any vector as in (A.1(α)), that is, Au = 0 and hα, viu + v ∈ Ln ∀v ∈ Ext(Ln ). In order to establish this result, it is enough to show that u ∈ Ker(A) ∩ Xα . Clearly u ∈ Ker(A). To see that u ∈ Xα , let v 0 be an arbitrary vector in Vα . Since v 0 ∈ Ext(Ln ) and u is as in (A.1(α)), we have hα, v 0 iu + v 0 ∈ Ln . Then hα, v 0 i = 1 implies u ∈ Ln − v 0 /hα, v 0 i = Ln − v 0 implying that u ∈ Xα . u t

We next show that in fact Xα ⊆ Ln ; and thus condition (3) is already covered by condition (A.0). Because condition (A.0) is a consequence of condition (A.1), this proves their equivalence. As a result, the definition of Ln -sublinearity is significantly simplified. Proposition 6 When K = Ln where n ≥ 3, any valid inequality that satisfies conditions (A.0) and (A.2) is K-sublinear. Proof To establish this result, given any α ∈ Ext(Ln ), we will prove that Xα ⊆ Ln . Claim If x ∈ Xα , then xn ≥ 0. Proof of Claim: Consider the n − 2 dimensional subspace {x ∈ Rn : xT α = 0, xn = 0}, and let u ˆ be any nonzero vector this subspace. Without loss of generality we may assume Pfrom 1 2 that u ˆ is scaled such that n− two vectors u, w by setting u ˆ i=1 i = 1/αn . Next, we construct P 1 2 ui = u ˆi and wi = −u ˆi for i ∈ {1, . . . , n − 1} and un = wn = in− ˆi . Then by construction =1 u u, w ∈ Ext(Ln ), αT u = αT w = 1, i.e., u, w ∈ Vα , and ui = −wi for i ∈ {1, . . . , n− 1}, un = wn . Because u, w ∈ Vα , Xα ⊆ (Ln − u) ∩ (Ln − w). Next, we show that if x ∈ Xα ⊆ (Ln − u) ∩ (Ln − w), we must have xn ≥ 0. Note that Ln − v =



x ∈ Rn :

n− X1

 (xi + vi )2 ≤ (xn + vn )2 , xn + vn ≥ 0 .

i=1

Therefore, x satisfies xn + un = xn + wn ≥ 0 as well as the following two conditions: n− X1

n− X1 i=1

(xi − ui )2 =

i=1 n− X1

(xi + ui )2 ≤ (xn + un )2 ,

(xi + wi )2 ≤ (xn + wn )2 = (xn + un )2 .

i=1

On Sublinear Inequalities for Mixed Integer Conic Programs

13

These two inequalities lead to: (n−1 ) n− X X1 2 2 (xn + un ) ≥ max (xi + ui ) , (x i − u i ) 2

=

n− X1

2

i=1 n− X1

xi +

i=1

i=1 (n−1 X

2

ui + max

i=1

2xi ui , −

i=1

n− X1

) 2xi u i

.

i=1

{z

|

}

≥0

n 2 Because Pn−1 2 u ∈ Ext(L ), the above inequality implies that xn + 2xn un = xn (xn + 2un ) ≥ i=1 xi ≥ 0. Also, 0 ≤ xn + un < xn + 2un since un > 0. Thus, xn (xn + 2un ) ≥ 0 can hold only if xn ≥ 0, which establishes our claim that xn ≥ 0. u t

The nonnegativity of xn implies xn + vn ≥ 0 for any v ∈ Vα . Therefore, for x with xn ≥ 0, x ∈ Xα if and only if: n− X1

2 x2i − x2n ≤ vn −

i=1

n− X1

vi2 −2

 n− X1

i=1

|

{z

 xj vj − xn vn

∀v ∈ Vα .

(4)

j =1

}

=0 ∀v∈Vα

Moreover, using the definition of Vα , condition (4) can be rewritten as n− X1

x2i − x2n ≤ 2 infn v∈R

i=1

 xn vn −

n− X1

2 xj vj : αT v = 1, vn =

n− X1



vi2 , vn ≥ 0 .

i=1

j =1

The right-hand side of this inequality involves optimizing a linear function over a nonconvex set. Nevertheless, note that     n− n− X1 2 X1 2 2 conv v ∈ Rn : αT v = 1, vn2 = vi , vn ≥ 0 = v ∈ Rn : αT v = 1, vn ≥ vi , vn ≥ 0 i=1

i=1

n

n

T

n

= v ∈ R : α v = 1, v ∈ L

o

.

Moreover, optimizing a linear function over a set is equivalent to optimizing it over its closed convex hull. Therefore, we arrive at the following inequality with a conic optimization problem n− X1 i=1

x2i − x2n ≤ 2 infn v∈R

 xn vn −

n− X1



xj vj : αT v = 1, v ∈ Ln .

j =1

α Note that the above conic optimization problem is bounded below; and v = kαk is a feasible 2 solution to it. Next, we show that this problem is strictly feasible. Because α is an extreme ray of Ln , there exists a unique ray γ ∈ Ln such that αT γ = 0, that is, γi = −αi for i = 1, . . . , n − 1 and γn = αn . Let v be any feasible solution to this optimization problem. Then v + γ is also feasible because v and γ are both in the cone and are not collinear and αT (v + γ ) = αT v = 1. Moreover, since Ln is strictly convex, summing up any two distinct extreme rays from Ln results in an interior

14

Fatma Kılın¸c-Karzan, Daniel E. Steffy

point of Ln . Thus, v + γ is a strictly feasible solution to the above optimization problem. Therefore, the strong conic duality theorem [8] together with Ln = K = K∗ implies     −x1 − α1 τ       n−     .. X1 2   2 n . xi − xn ≤ 2 sup τ :  ∈L   −xn−1 − αn−1 τ  τ ∈R    i=1     xn − αn τ

( = 2 sup τ :

n− X1

τ ∈R

 = 2 sup τ : τ ∈R

) 2

2

(−xi − αi τ ) ≤ (xn − αn τ ) , xn − αn τ ≥ 0

i=1 n− X1

x2i − x2n + 2(αT x)τ +

 n− X1

i=1

2 αi2 − αn



τ 2 ≤ 0, τ ≤

i=1

|

{z

xn xi , τ ≤ αn

)

xn αn



}

=0 as α∈Ext(Ln )

( T

2

= 2 sup τ : 2(α x)τ ≤ xn − τ ∈R

n− X1 i=1

2

.

In order to finish the proof, it is sufficient to show that the above requirement is equivalent to x ∈ ±Ln . We will consider the following two cases separately: Case 1: αT x > 0.

In this case we have n− X1

( 2

2

xi − xn ≤ min

equivalently

P n−1 i=1

2

2

xi − xn

1+

|



1

(αT x) {z }

x2i − x2n −(αT x)

)

i=1

,

αn

i=1



Pn−1

2xn

Pn−1 ≤

x2i − x2n , −(αT x)

i=1

≤ 0, which holds if and only if x ∈ ±Ln .

>0

Case 2: αT x ≤ 0.

In this case we establish the result by showing that the optimization problem above is feasible only when x ∈ Ln . Using the nonpositivity of αT x, we rewrite the constraints of the optimization problem as n− X1 2 xn , xi − x2n ≤ −2(αT x)τ ≤ −2(αT x) αn

i=1

which can be rearranged as n− X1

x2i − x2n + 2x2n +

i=1 2 Because α ∈ Ext(Ln ), we have αn =

0≥

n− X1 

x2i +

i=1

This is only possible if xi =

−αi αn xn

Pn−1 i=1

2xn αn

n−1 2xn X

αn

αi xi ≤ 0.

i=1

αi2 , leading to:

αi xi +

αi2 2 x 2 n αn

 =

n− X1 

xi +

i=1

∀i = 1, ..n − 1 and thus: n− X1 i=1

x2i =



x2n 2 αn

 n− X1 i=1

αi2 = x2n ,

αi xn αn

2

.

On Sublinear Inequalities for Mixed Integer Conic Programs

15

which holds if and only if x ∈ ±∂Ln . Finally, combining x ∈ ±∂Ln with xn ≥ 0, we conclude that x ∈ Ln .

u t

Lemma 2 and Proposition 6 imply that for K = Ln with n ≥ 3 conditions (A.0) and (A.1) are equivalent; and hence any valid inequality satisfying conditions (A.0) and (A.2) is Ln -sublinear. In particular, by [21, Proposition 6], we arrive at the following corollary. Corollary 3 When n ≥ 3, any valid inequality for S (A, Ln , B) is Ln -sublinear.

Acknowledgments

The authors wish to thank the review team for their constructive feedback that improved the presentation of the material in this paper. The research of the first author is supported in part by NSF grant CMMI 1454548.

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